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## Planned Motion Equations of Free-running Grain Mixture Flow

### Abstract

In the article the equations have been worked making it possible to model the motion of freerunning grain mixture flow on a flat sloping vibrating sieve within the framework of shallow water theory. Free-running grain mixture is considered as a heterogeneous system consisting of two phases, one of which represents solid particles and the other one gas. The mixture is brought into a state of fluidity by means of high-frequency vibration imposition. Coefficients of internal and external friction and dynamic-viscosity decrease by exponential law as the fluctuation intensity is increased. When considering grain mixture dynamics, the following assumptions are put forward: we ignore the air presence in space between particles, we consider the density of particles to be constant, the free-running mixture is similar to Newtonian liquid. The basic system of equations of grain mixture dynamics is due to the laws of continuum mechanics. The equation of continuity is issued from the law of conservation of mass, and the dynamic equations are issued from the law of variation of momentum. The stress tensor equals to the sum of the equilibrium tensor and the dissipative tensor. The equilibrium part of the stress tensor is represented by the spherical tensor, which is found to conform to Pascal law for liquids, and the dissipative part, which is responsible for viscous force effect and defined by Navier-Stokes law. Boundary conditions on the surfaces (restricting the capacity of the free-running grain mixture) have been researched. The distributions of apparent density and velocity field are assigned at the inlet and outlet flow sections of the mixture. The normal velocity component of the grain mixture on the side frames and on the sieve becomes zero, which meets the no-fluid-loss condition of the medium through the frame. Beyond that point at this time we satisfy dynamic conditions, which characterize the mixture sliding down the hard frame, motion flow resistance force is represented as average velocity linear dependence. A kinematic condition and two dynamic ones are stipulated on the free surface layer. One of the conditions states mass flow continuity across the free surface, the other one states the stress continuity while passing through the free surface. The basic premise of planned motion equations is the condition of small size of flow depth in comparison with its width. With the use of shallow water theory the basic principles of the equations of flow dynamics are simplified and for their solving a Cauchy problem can be set.
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